To me, flatting the 3-bet PRE was bad. I'd work toward having a 4-bet range for spots like this, which is a reasonably common spot.

Your flat call prices in the UTG min-raiser to call, which significantly weakens your chances of winning with AA. You want to get it heads-up ASAP with AA.

Other than that, calling the flop is pretty standard. You have the deck crippled, and can only be beat by straight flush, so letting Villain set the price is fine.

OTT, you should be betting. You have the nuts. You put chips in the pot when you have the nuts.
You simply must bet the top of your range.
If opponents fold too much when you bet, then add the bottom of your range to that, too.
Letting him set the price and calling behind is fine as long as the price isn't laughably low.
The guiding principle here is that you want to bet as much as he'll call, and set up a river shove that is less than a pot-sized bet.

OTR you shorted yourself ~40 BB by not getting your whole stack in by the end. Your overbet may have been max value at that point, which demonstrates why the turn bet which sets up your river sizing is so important.